For the computability of subsets of real numbers, several reasonable notions have been suggested in the literature. We compare these notions in a systematic way by relating them to pairs of ‘basic’ ones. They turn out to coincide for full-dimensional convex sets; but on the more general class of regular sets, they reveal rather interesting ‘weaker/stronger’ relations. This is in contrast to single real numbers and vectors where all ‘reasonable’ notions coincide.

Humans have long been characterized as poor probabilistic reasoners when presented with explicit numerical information. Bayesian word problems provide a well-known example of this, where even highly educated and cognitively skilled individuals fail to adhere to mathematical norms. It is widely agreed that natural frequencies can facilitate Bayesian reasoning relative to normalized formats (e.g. probabilities, percentages), both by clarifying logical set-subset relations and by simplifying numerical calculations. Nevertheless, between-study performance on “transparent” Bayesian problems varies widely, and generally remains (...) rather unimpressive. We suggest there has been an over-focus on this representational facilitator (i.e. transparent problem structures) at the expense of the specific logical and numerical processing requirements and the corresponding individual abilities and skills necessary for providing Bayesian-like output given specific verbal and numerical input. We further suggest that understanding this task-individual pair could benefit from considerations from the literature on mathematical cognition, which emphasizes text comprehension and problem solving, along with contributions of online executive working memory, metacognitive regulation, and relevant stored knowledge and skills. We conclude by offering avenues for future research aimed at identifying the stages in problem solving at which correct versus incorrect reasoners depart, and how individual difference might influence this time point. (shrink)

As an introduction to our work, we emphasize the parallel interpretation of abstract tools and the concepts of undetermined and vague information. Imprecision, uncertainty and their relationships are inspected. Suitable interpretations of the fuzzy sets theory are applied to legal phenomena in an attempt to clearly circumscribe the possible applications of the theory. The fundamental notion of reference sets is examined in detail, hence highlighting their importance. A systematic and combinatorial classification of the relevant subsets of the legal field is (...) supplied for practical application. Although the use of the fuzzy sets theory is sometimes suggested as a palliative measure (no competition exists), it can also be complementary (serve as a building block to improve modelisation). An Appendix gives a brief recall of the key-concepts of the axiomatic theory of fuzziness and its developments: fuzzy sets, fuzzy logic, fuzzy control and theory of possibility. (shrink)

We generalize Moss and Parikh's logic of knowledge, effort, and topological reasoning, in two ways. We develop both a multi-agent and a multi-method setting for it. In each of these cases, we prove a corresponding soundness and completeness theorem, and we show that the new logics are decidable. Our methods of proof rely on those for the original system. This might have been expected, since that system is conservatively extended for the given situation. Several technical details are different nevertheless (...) here. (shrink)

Subset Spaces were introduced by L. Moss and R. Parikh in [8]. These spaces model the reasoning about knowledge of changing states.In [2] a kind of subset space called intersection space was considered and the question about the existence of a set of axioms that is complete for the logic of intersection spaces was addressed. In [9] the first author introduced the class of directed spaces and proved that any set of axioms for directed frames also characterizes (...) intersection spaces.We give here a complete axiomatization for directed spaces. We also show that it is not possible to reduce this set of axioms to a finite set. (shrink)

In this study, we claim that political liberalism, despite harsh criticism, is still the best option available for providing a just and stable society. However, we maintain that political liberalism needs to be revised so as to be justifiable from the perspective of not only the “reasonable” in a Rawlsian sense (that we define as “fully” reasonable) but also the ones whom Rawls labels as “unreasonable.” To support our claim, going beyond Rawls’s original account, we unpack the concept of unreasonableness (...) and identify three different subsets that we label as the “partially reasonable,” the “non-reasonable,” and the “unreasonable.” We argue that both the “fully” reasonable and the “partially reasonable” would be included into the constituency of public justification; more specifically, we claim that the latter would support liberal institutions out of their reasons: we define these reasons as mutually intelligible reasons and claim that they allow to acknowledge the importance of a convergence approach to public justification. As for the “non-reasonable” and “unreasonable,” we claim that they cannot be included in the constituency of public justification, but they nonetheless could be compliant with liberal institutions if political liberalism offers them some reasons to comply: here, we claim that political liberalism should include them through engagement and propose reasoning from conjecture as an effecting way of offering reasons for compliance. In particular, we claim that through reasoning from conjecture, the “non-reasonable” could find conciliatory reasons to comply with liberal institutions on a stable base. With regard to the “unreasonable” in the strict sense, we claim that through reasoning from conjecture, their unreasonableness could be contained and they could find reasons—even if just self-interested—for complying with liberal institutions rather than defying them. In our discussion, we consider the different subsets not as “frozen” but as dynamic and open to change, and we aim to propose a more complex and multilayered approach to inclusion that would be able to include a wider set of people. To strengthen our argument, we show that the need for a wider public justification and for broader inclusion in liberal societies is grounded in respect for persons both as equal persons and as particular individuals. In particular, we claim that individuals’ values, ends, commitments, and affiliations activate demands of respect and can strengthen the commitment to the liberal–democratic order. Through a reformulation of the role of respect in liberal societies, we also show a kind of social and communitarian dimension that, we claim, is fully compatible with political liberalism and opens it up to “civic friendship” and “social solidarity,” which are constitutive elements for the development of a sense of justice and for the realization of a just and stable society. (shrink)

We present a representation for reasoning and planning with an incomplete state description (open-world) called PSIPLAN-S. The presented formalism has several properties critical for application domains with a large degree of incompleteness in the state description, particularly, in domains with a large or unknown set of all objects. The formalism offers (1) considerably expressive state and goal description language, that includes limited universal quantification, (2) representation of sensing actions and knowledge goals, (3) a correct and complete state update procedure, (...) and (4) complete reasoning within a substantial subset of the language. The approach is illustrated by examples from a working system. (shrink)

A similarity relation is a reflexive and symmetric binary relation between objects. Similarity is relative: it depends on the set of properties of objects used in determining their similarity or dissimilarity. A multi-modal logical language for reasoning about relative similarities is presented. The modalities correspond semantically to the upper and lower approximations of a set of objects by similarity relations corresponding to all subsets of a given set of properties of objects. A complete deduction system for the language is (...) presented. (shrink)

We present a bimodal logic suitable for formalizing reasoning about points and sets, and also states of the world and views about them. The most natural interpretation of the logic is in subset spaces , and we obtain complete axiomatizations for the sentences which hold in these interpretations. In addition, we axiomatize the validities of the smaller class of topological spaces in a system we call topologic . We also prove decidability for these two systems. Our results on (...) topologic relate early work of McKinsey on topological interpretations of S4 with recent work of Georgatos on topologic. Some of the results of this paper were presented at the 1992 conference on Theoretical Aspects of Reasoning about Knowledge. (shrink)

Suppose that we want to frame a conception of reasons that isn't relativized to the inclinations of particular agents. That is, we want to identify particular things that count as reasons for acting simpliciter and not merely as reasons for some agents rather than others, depending on their inclinations. One way to frame such a conception is to name some features that an action can have and to say that they count as reasons for someone whether or not he is (...) inclined to care about them. The problem with the resulting conception, as we have seen, is that it entails the normative judgment that one ought to be inclined to care about the specified features, on pain of irrationality, and this normative judgment requires justification. The advantage of internalism is that it avoids these normative commitments. It says that things count as reasons for someone only if he is inclined to care about them, and so it leaves the normative question of whether to care about them entirely open. Yet if we try to leave this question open, by defining things as reasons only for those inclined to care about them, we'll end up with a definition that's relativized to the inclinations of particular agents—won't we? Not necessarily. For suppose that all reasons for acting are features of a single kind, whose influence depends on a single inclination. And suppose that the inclination on which the influence of reasons depends is, not an inclination that distinguishes some agents from others, but rather an inclination that distinguishes agents from nonagents. In that case, to say that these features count as reasons only for those who are inclined to care about them will be to say that they count as reasons only for agents—which will be to say no less than that they are reasons for acting, period, since applying only to agents is already part of the concept of reasons for acting. The restriction on the application of reasons will drop away from our definition, since it restricts their application, not to some proper subset of agents, but rather to the set of all agents, which is simply the universe of application for reasons to act. (shrink)

Philosophy & Social Criticism, Volume 48, Issue 9, Page 1226-1246, November 2022. In this study, we claim that political liberalism, despite harsh criticism, is still the best option available for providing a just and stable society. However, we maintain that political liberalism needs to be revised so as to be justifiable from the perspective of not only the “reasonable” in a Rawlsian sense but also the ones whom Rawls labels as “unreasonable.” To support our claim, going beyond Rawls’s original account, (...) we unpack the concept of unreasonableness and identify three different subsets that we label as the “partially reasonable,” the “non-reasonable,” and the “unreasonable.” We argue that both the “fully” reasonable and the “partially reasonable” would be included into the constituency of public justification; more specifically, we claim that the latter would support liberal institutions out of their reasons: we define these reasons as mutually intelligible reasons and claim that they allow to acknowledge the importance of a convergence approach to public justification. As for the “non-reasonable” and “unreasonable,” we claim that they cannot be included in the constituency of public justification, but they nonetheless could be compliant with liberal institutions if political liberalism offers them some reasons to comply: here, we claim that political liberalism should include them through engagement and propose reasoning from conjecture as an effecting way of offering reasons for compliance. In particular, we claim that through reasoning from conjecture, the “non-reasonable” could find conciliatory reasons to comply with liberal institutions on a stable base. With regard to the “unreasonable” in the strict sense, we claim that through reasoning from conjecture, their unreasonableness could be contained and they could find reasons—even if just self-interested—for complying with liberal institutions rather than defying them. In our discussion, we consider the different subsets not as “frozen” but as dynamic and open to change, and we aim to propose a more complex and multilayered approach to inclusion that would be able to include a wider set of people. To strengthen our argument, we show that the need for a wider public justification and for broader inclusion in liberal societies is grounded in respect for persons both as equal persons and as particular individuals. In particular, we claim that individuals’ values, ends, commitments, and affiliations activate demands of respect and can strengthen the commitment to the liberal–democratic order. Through a reformulation of the role of respect in liberal societies, we also show a kind of social and communitarian dimension that, we claim, is fully compatible with political liberalism and opens it up to “civic friendship” and “social solidarity,” which are constitutive elements for the development of a sense of justice and for the realization of a just and stable society. (shrink)

It has long been recognized that the concept of inconsistency is a central part of commonsense reasoning. In this issue, a number of authors have explored the idea of reasoning with maximal consistent subsets of an inconsistent stratified knowledge base. This paradigm, often called “coherent-based reasoning", has resulted in some interesting proposals for para-consistent reasoning, non-monotonic reasoning, and argumentation systems. Unfortunately, coherent-based reasoning is computationally very expensive. This paper harnesses the approach of approximate entailment (...) by Schaerf and Cadoli [SCH 95] to develop the concept of “approximate coherent-based reasoning". To this end, we begin to present a multi-modal propositional logic that incorporates two dual families of modalities: □S and?S defined for each subset S of the set of atomic propositions. The resource parameter S indicates what atoms are taken into account when evaluating formulas. Next, we define resource-bounded consolidation operations that limit and control the generation of maximal consistent subsets of a stratified knowledge base. Then, we present counterparts to existential, universal, and argumentative inference that are prominent in coherence-based approaches. By virtue of modalities □S and?S, these inferences are approximated from below and from above, in an incremental fashion. Based on these features, we show that an anytime view of coherent-based reasoning is tenable. (shrink)

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the (...) failure of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares. (shrink)

We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, in (...) the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning. (shrink)

According to probabilistic theories of reasoning in psychology, people's degree of belief in an indicative conditional `if A, then B' is given by the conditional probability, P(B|A). The role of language pragmatics is relatively unexplored in the new probabilistic paradigm. We investigated how consequent relevance a ects participants' degrees of belief in conditionals about a randomly chosen card. The set of events referred to by the consequent was either a strict superset or a strict subset of the set (...) of events referred to by the antecedent. We manipulated whether the superset was expressed using a disjunction or a hypernym. We also manipulated the source of the dependency, whether in long-term memory or in the stimulus. For subset-consequent conditionals, patterns of responses were mostly conditional probability followed by conjunction. For superset-consequent conditionals, conditional probability responses were most common for hypernym dependencies and least common for disjunction dependencies, which were replaced with responses indicating inferred consequent irrelevance. Conditional probability responses were also more common for knowledge-based than stimulus-based dependencies. We suggest. (shrink)

This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).

Relational reasoning is a complex form of human cognition involving the evaluation of relations between mental representations of information. Prior studies have modified stimulus properties of relational reasoning problems and examined differences in difficulty between different problem types. While subsets of these stimulus properties have been addressed in separate studies, there has not been a comprehensive study, to our knowledge, which investigates all of these properties in the same set of stimuli. This investigative gap has resulted in different (...) findings across studies which vary in task design, making it challenging to determine what stimulus properties make relational reasoning—and the putative formation of mental models underlying reasoning—difficult. In this article, we present the Multidimensional Relational Reasoning Task, a task which systematically varied an array of stimulus properties within a single set of relational reasoning problems. Using a mixed-effects framework, we demonstrate that reasoning problems containing a greater number of the premises as well as multidimensional relations led to greater task difficulty. The MRRT has been made publicly available for use in future research, along with normative data regarding the relative difficulty of each problem. (shrink)

We extend Moss and Parikh’s bi-modal system for knowledge and effort by means of hybrid logic. In this way, some additional concepts from topology related to knowledge can be captured. We prove the soundness and completeness as well as the decidability of the extended system. Special emphasis will be placed on algebras.

then E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map ν be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be (...) constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory3, the following.. (shrink)

This paper first provides a brief survey of a possibilistic handling of default rules. A set of default rules of the form, "generally, from α deduce β", is viewed as the family of possibility distributions satisfying constraints expressing that the situation where α and β is true has a greater plausibility than the one where α and ⇁β is true. When considering only the subset of linear possibility distributions, the well-known System P of postulates proposed by Kraus, Lehmann and (...) Magidor, has been obtained. We also present two rational extensions: one based on the minimum specificity principle and the other is based on the lexicographic ordering. The second part of the paper presents an empirical study of three desirable properties for a consequence relation that capture default reasoning: Rationality, Property Inheritance and Ambiguity Preservation. An experiment is conducted to investigate 13 patterns of inference for the test of these properties. Our experimental apparatus confirms previous results on the relevance of System P, and enforces the psychological relevance of the studied properties. (shrink)

We study a notion of ‘o-amorphous’ which bears the same relationship to ‘o-minimal’ as ‘amorphous’ 191–233) does to ‘strongly minimal’. A linearly ordered set is said to be o-amorphous if its only subsets are finite unions of intervals. This turns out to be a relatively straightforward case, and we can provide a complete ‘classification’, subject to the same provisos as in Truss . The reason is that since o-amorphous is an essentially second-order notion, it corresponds more accurately to 0-categorical o-minimal, (...) and our classification is thus very similar to the one given in 565–592) for that case. More interesting structures arise if we replace ‘interval’ in the definition by ‘convex set’, giving us the class of weakly o-amorphous sets. Here, in fact, there are so many examples that a complete classification seems out of the question. We illustrate some of the structures which these may exhibit, and classify them in certain instances not too far removed from the o-amorphous case. (shrink)

ABSTRACT Two objects o1, o2 of an information system are said to be complementary with respect to attribute a if α(o1) = -α(o2), where α(o) is the set of values of attribute a assigned to o. They are said to be complementary with respect to a set of attributes A if they are complementary with respect to each attribute α ε A. A multi-modal logical language for reasoning about complementarity relations is presented, with modalities [A] and ?A? parameterised by (...) subsets of a given set ATTR of attributes. Two complete deduction system for the language are presented: a Rasiowa-Sikorski style system using signed formulae, and an equivalent sequent calculus system in Gentzen style complete for theories. (shrink)

The classical measures of voting power are based on players' decisiveness or full causal efficacy in vote configurations or divisions. We design an alternative, recursive measure departing from this classical approach. We motivate the measure via an axiomatic characterisation based on reasonable axioms and by offering two complementary interpretations of its meaning: first, we interpret the measure to represent, not the player's probability of being decisive in a voting structure, but its expected probability of being decisive in a uniform random (...) walk from a vote configuration in the subset lattice (through which we represent the voting structure); and, second, we interpret it as representing a player's expected efficacy, thereby incorporating the notion of partial and not just full causal efficacy. We shore up our measure by demonstrating that it satisfies a set of postulates any reasonable voting measure should satisfy, namely, the iso-invariance, dummy, dominance, donation, minimum-power bloc, and quarrel postulates. (shrink)

Based on a close study of benchmark examples in default reasoning, such as Nixon Diamond, Penguin Principle, etc., this paper provides an in depth analysis of the basic features of default reasoning. We formalize default inferences based on Modus Ponens for Default Implication, and mark the distinction between "local inferences"(to infer a conclusion from a subset of given premises) and "global inferences"(to infer a conclusion from the entire set of given premises). These conceptual analyses are captured by (...) a formal semantics that is built upon the set-selection function technique. A minimal logic system M of default reasoning that accommodates Modus Ponens for Default Implication and suitable for local inferences is proposed, and its soundness is proved. (shrink)

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a (...) fuzzy subset of [0, 1].Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc. (shrink)

We report two studies on the effect of implicitly versus explicitly conveying affirmation and denial problems about conditionals. Recently Evans and Handley (1999) and Schroyens et al. (1999b, 2000b) showed that implicit referencing elicits matching bias: Fewer determinate inferences are made, when the categorical premise (e.g., B) mismatches the conditional's referred clause (e.g., A). Also, the effect of implicit affirmation (B affirms not-A) is larger than the effect of implicit denial (B denies A). Schroyens et al. hypothesised that this interaction (...) is due to uncertainty in the case-wise affirmation of the contrast class of negated elements involved in implicit affirmations. In Experiment 1 we tested this hypothesis by manipulating the set size of the conditional clauses. The results confirm that binary sets, where the contrast class is a singleton, eliminate the differential effect of implicit affirmation and denial. With non-binary sets the interaction is not modulated by the scope of the contrast class (3, 5, 9 elements). Experiment 2 further investigated the role of contrast classes by using class inclusion to construct implicit affirmations (Mammal vs Mammal or Monkey) and implicit denial (No-Mammal vs Mammal or Monkey), in addition to the standard implicit problems mediated by contrast-class inclusion [(No-)Mammal/No-Mammal; Reptile; Snake). Findings indicate that class inclusion (Mammal/Monkey; Reptile/ Snake) only marginally affects performance, and is independent of the type of problem. This would suggest that the implicitness problem-type interaction is dependent on constructing contrast classes. However, the experiment failed to replicate the interaction, even on the subset of problems repeating the abstract letter/number format of Experiment 1. Moreover, with the natural binary set-sizes (vowels/consonants) the implicitness effect was eliminated entirely. (shrink)

Language employs various coordinators to connect propositions, a subset of which are “logical” in nature and thus analogous to the truth operators of formal logic. We here focus on two linguistic connectives and their negations: conjunction _and_ and (inclusive) disjunction _or_. Linguistic connectives exhibit a truth-conditional component as part of their meaning (their semantics), but their use in context can give rise to various implicatures and presuppositions (the domain of pragmatics) as well as to inferences that go beyond semantic/pragmatic (...) properties (the result of reasoning processes). We provide a comprehensive review of the role of the logical connectives in language and argue that three sets of factors—semantic, pragmatic, and those related to reasoning—are separate and separable, though some details may differ cross-linguistically. As a way to showcase the argument, we present two experiments in language comprehension in Spanish wherein pragmatic content was minimised and reasoning processes neutered, thus potentially highlighting what might be the default meanings of the connectives under study. In Experiment 1 we show that the conjunctive reading of inclusive disjunction is available in positive contexts other than in syntactically intricate cases such as downward entailing and free choice contexts, contrary to what has been claimed in the literature. In Experiment 2 we show that negated conjunctions and disjunctions in Spanish can easily receive the same interpretation when contrasted against the same context and, moreover, that these interpretations match those available in English, despite claims from the literature that linguistic connectives and local negation interact differently in English and Romance languages. (shrink)

A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often forcing constructions that add subsets to models are cited as evidence in favour of the latter. This paper informs this debate by analysing ways the Universist might interpret this discourse that seems (...) to necessitate the addition of subsets to V. We argue that despite the prima facie incoherence of such talk for the Universist, she nonetheless has reason to try and provide interpretation of this discourse. We analyse extant interpretations of such talk, and analyse various tradeoffs in naturality that might be made. We conclude that the Universist has promising options for interpreting different forcing constructions. (shrink)

Recent work in English speaking moral philosophy has seen the rise to prominence of the idea of a normative reason1. By ‘normative reasons’ I mean the reasons agents appeal to in making rational claims on each other. Normative reasons are good reasons on which agents ought to act, even if they are not actually motivated accordingly2. To this extent, normative reasons are distinguishable from the motivating reasons agents appeal to in reason explanations. Even agents who fail to act on their (...) normative reasons can be said to act on reasons insofar as their actions are rationally intelligible. Thus, when it is said that agents may never use violence in self-defence, this is naturally interpreted to mean that there are powerful normative reasons not to use violence even in selfdefence, even though some agents would use violence in selfdefence. Normative reasons are reasons to pursue ends, where by ends I mean a subset of objects of possible desire, such as taking a stroll or giving all your money to charity. The set of objects of possible desire might include items that are not straightforwardly ends of action. For example, you might want the world to be a better place, or want a secure basis in knowledge of relevant facts to be assigned the highest priority in the assessment of people’s preferences. Objects of possible desire are a subset of objects of possible response, where by ‘response’ I mean the whole range of prepositional attitudes, including desires, preferences, beliefs, commitments and so on. I use the term ‘option’ to refer to objects of possible response in this wider sense. Recent philosophical claims about the grounds of normative reasons can be divided into two strands. Each strand takes as its starting point what is perceived to be a fundamental constraint embodied in normative reason attributions.. (shrink)

. This paper first provides a brief survey of a possibilistic handling of default rules. A set of default rules of the form, “generally, from α deduce β”, is viewed as the family of possibility distributions satisfying constraints expressing that the situation where α and β is true has a greater plausibility than the one where a and - β is true. When considering only the subset of linear possibility distributions, the well-known System P of postulates proposed by Kraus, (...) Lehmann and Magidor, has been obtained. We also present two rational extensions: one based on the minimum specificity principle and the other is based on the lexicographic ordering. The second part of the paper presents an empirical study of three desirable properties for a consequence relation that capture default reasoning: Rationality, Property Inheritance and Ambiguity Preservation. An experiment is conducted to investigate 13 patterns of inference for the test of these properties. Our experimental apparatus confirms previous results on the relevance of System P, and enforces the psychological relevance of the studied properties. (shrink)

This paper presents a uniform semantic treatment of nonmonotonic inference operations that allow for inferences from infinite sets of premises. The semantics is formulated in terms of selection functions and is a generalization of the preferential semantics of Shoham (1987), (1988), Kraus, Lehman, and Magidor (1990) and Makinson (1989), (1993). A selection function picks out from a given set of possible states (worlds, situations, models) a subset consisting of those states that are, in some sense, the most preferred ones. (...) A proposition α is a nonmonotonic consequence of a set of propositions Γ iff α holds in all the most preferred Γ-states. In the literature on revealed preference theory, there are a number of well-known theorems concerning the representability of selection functions, satisfying certain properties, in terms of underlying preference relations. Such theorems are utilized here to give corresponding representation theorems for nonmonotonic inference operations. At the end of the paper, the connection between nonmonotonic inference and belief revision, in the sense of Alchourrón, Gärdenfors, and Makinson, is explored. In this connection, infinitary belief revision operations that allow for the revision of a theory with a possibly infinite set of propositions are introduced and characterized axiomatically. (shrink)

People often struggle with Bayesian reasoning. However, research showed that people’s performance (and rationality) can be supported by the way of representing the statistical information. First, research showed that using natural frequencies instead of probabilities as format of statistical information increases people’s performance in Bayesian situations thoroughly. Second, research also yielded that people’s performance increases through using visualization. We build our paper on existing research in this field. The main aim is to analyse people’s strategies in Bayesian situations that (...) are still erroneous although statistical information is represented by natural frequencies and visualizations. In particular, we compare two pairs of visualization with similar numerical information (tree diagram vs. unit square and double-tree diagram vs. 2x2-table) concerning their impact on people’s erroneous strategies in Bayesian situations. For this aim, we conducted an experiment with 540 university students. The students were randomly assigned to four conditions that are defined by the four different visualizations of statistical information. The students were asked to indicate a fraction as response in four Bayesian situations. We documented the numerator and the denominator of the students’ responses representing a basic set and a subset in a Bayesian situation. Our results show that people’s erroneous strategies are highly dependent on the visualization. A central finding is that the visualization’s characteristic of making the nested-sets structure of a Bayesian situation transparent has a facilitating effect on people’s Bayesian reasoning. For example, compared to the unit square, a tree diagram does not explicitly visualize the set-subset relations that are relevant in a Bayesian situation. Accordingly, compared to a unit square, a tree diagram partly hinders people to find the correct denominator in a Bayesian situation, and, in particular, triggers to select a wrong numerator. By analyzing people’s erroneous strategies in Bayesian situations, we contribute to investigating approaches to facilitate Bayesian reasoning and to further develop the teaching of Bayesian reasoning. (shrink)

The scientific reasoning strategies used to discover a new concept in a scientific domain were investigated in two studies. An innovative task in which subjects discover new concepts in molecular biology was used. This task was based upon one set of experiments that Jacob and Monod used to discover how genes are controlled, and for which they were awarded the Nobel prize. In the two studies reported in this article, subjects were taught some basic facts and experimental techniques in (...) molecular biology, using a simulated molecular genetics laboratory on a computer. Following their initial training, they were then asked to discover how genes are controlled by other genes. In Study 1, subjects found no evidence that was consistent with their initial hypothesis. Subjects then set one of two goals for conducting experiments and evaluating data. One goal was to search for evidence consistent with the current hypothesis (and they did not attend to the features of discrepant findings); none of the subjects who only had this goal succeeded at discovering how the genes were controlled. Other subjects in Study 1 used a different goal: Upon noticing evidence inconsistent with their current hypothesis, these subjects set a new goal of attempting to explain the cause of the discrepant findings. Using this goal, a subset of these subjects discovered the correct solution to the problem. Study 2 was conducted to test the hypothesis that subjects' goals of finding evidence consistent with their current hypothesis blocks consideration of alternate hypotheses and generation of new goals, it was predicted that if subjects could achieve their initial goal of discovering evidence consistent with their current hypothesis, they would then attend to particular features of discrepant evidence and solve the problem. To test this prediction, an additional mechanism of genetic control that was consistent with subjects' initial goal was added to the genes. Here, subjects had to discover two mechanisms of control: one mechanism consistent with their current hypothesis, and one inconsistent with their hypothesis. Twice as many subjects reached the correct solution in Study 2 than in Study 1. The findings of the two studies indicate that goals provide a powerful constraint on the cognitive processes underlying scientific reasoning and that the types of goals that are represented determine many of the reasoning errors that subjects make. (shrink)

Inductive reasoning aims at constructing rules and models of general applicability from a restricted set of observations. Induction is a keystone in natural sciences, and it influences diverse application fields such as engineering, medicine and economics. More generally, induction plays a major role in the way humans learn and operate in their everyday life. The level of reliability that a model achieves depends on how informative the observations are relative to the flexibility of the process by which the model (...) is constructed. When the process is articulated so that the model can incorporate descriptive details and subtleties, a large set of informative observations are required to reliably tune the model, whereas models obtained from simple procedures can be tuned with fewer observations. This article introduces the concept of “dominance”, which refers to the situation in which a reduced subset of observations suffices to reconstruct the model. A mathematical framework is presented to quantify the reliability of learning procedures as a function of the size of the subset of dominant observations. Although limited in scope, we believe that our study can contribute to the understanding of some fundamental mechanisms by which knowledge is generated from observations in inductive reasoning. (shrink)

A region-based model of physical space is one in which the primitive spatial entities are regions, rather than points, and in which the primitive spatial relations take regions, rather than points, as their relata. Historically, the most intensively investigated region-based models are those whose primitive relations are topological in character; and the study of the topology of physical space from a region-based perspective has come to be called mereotopology. This paper concentrates on a mereotopological formalism originally introduced by Whitehead, which (...) employs a single primitive binary relation C(x, y) (read: "x is in contact with y"). Thus, in this formalism, all topological facts supervene on facts about contact. Because of its potential application to theories of qualitative spatial reasoning, Whitehead's primitive has recently been the subject of scrutiny from within the Artificial Intelligence community. Various results regarding the mereotopology of the Euclidean plane have been obtained, settling such issues as expressive power, axiomatization and the existence of alternative models. The contribution of the present paper is to extend some of these results to the mereotopology of three-dimensional Euclidean space. Specifically, we show that, in a first-order setting where variables range over tame subsets of ℝ³, Whitehead's primitive is maximally expressive for topological relations; and we deduce a corollary constraining the possible region-based models of the space we inhabit. (shrink)

Having, in previous papers, distinguished at least three forms of consciousness , I now further examine their differences. This examination has some surprising results. Having argued that neither C1 nor C2 is a phenomenological state?and so different from CN?I now show that CN itself is best thought of as a subclass of a larger state . CS is the set of image?representation states. CN is that set of CS states that we are also C2 about. I argue that CN states (...) should be considered a subset of CS states because they share with other CS states imagistic representational powers as opposed to the sentential ones of C1 and C2. And I argue that there are good reasons to consider any state which utilises imagistic representation as a sensation. I then show that the notion of an unconscious sensation makes sense if we think of it as a CS state that is not C2. More radically, I conclude that either it makes sense to say that there are unfelt sensations or there are felt sensations we are not conscious about. While there are good reasons for saying one or the other of these surprising things, I argue that the second, more radical sounding, alternative is actually better motivated?and not quite so surprising as it sounds. (shrink)

The multiple sources and functions of Heraclitus in Nietzsche’s writings should not be underestimated. Nietzsche’s early readings of Heraclitus are steeped in the Greek fragments, the doxographical tradition, and in philological scholarship. Hence, they are largely either fair interpretations of the extant fragments, clear translations of a select group of fragments into his own language, or improvisations based in part on a narrow subset of the spurious remarks set down in the doxographical tradition. Nietzsche’s later departures from this tradition (...) articulate an anti-metaphysical Heraclitus that he found in the Heraclitean fragments, which he takes to prefigure and later parrot his doctrines of becoming, the child, strife, chance, and eternal recurrence. For this reason, Nietzsche’s Heraclitus is necessarily no objective, individual historical person, but at various times and to various degrees a copy, an appropriative invention, and a personal-philological-philosophical archetype filtered through the lenses of ancient Greek understandings of how he must have lived, given what he said and what philosophy is; nineteenth-century German philological reconstructions of Heraclitean thought; and Nietzsche’s own need for such a precursor to voice his own doctrines and thereby lend them heft against the Platonic tradition of metaphysical philosophy. (shrink)

The requirements of instrumental rationality are often thought to be normative conditions on choice or intention, but this is a mistake. Instrumental rationality is best understood as a requirement of coherence on an agent's non-instrumental desires and means-end beliefs. Since only a subset of an agent's means-end beliefs concern possible actions, the connection with intention is thus more oblique. This requirement of coherence can be satisfied either locally or more globally, it may be only one among a number of (...) such requirements on an agent's total set of desires and beliefs, and it has no special connection with reasoning. An appreciation of these facts leads to a better understanding of both the nature and the significance of instrumental rationality. (shrink)

Abstract“Natural syllogisms” are arguments formally identifiable with categorical syllogisms that have an implicit universal affirmative premise retrieved from semantic memory rather than explicitly stated. Previous studies with adult participants (Politzer, 2011) have shown that the rate of success is remarkably high. Because their resolution requires only the use of a simple strategy (known as ecthesis in classic logic) and an operational use of the concept of inclusion (the recognition that an element that belongs to a subset must belong to (...) the set but not vice versa), it was hypothesized that these syllogisms would be within the grasp of non‐adult participants, provided they have acquired the notion of deductive validity. Here, 11‐year‐old children were presented with natural syllogisms embedded in short dialogs. The first experiment showed that their performance was equivalent to adults' highest level of performance in standard experiments on syllogisms. The second experiment, while confirming children's proficiency in solving natural syllogisms, showed that they outperformed children who solved non‐natural matched syllogisms in the same experimental setting. The results are also in agreement with the argumentation theory of reasoning. (shrink)

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets.

This paper presents a calculus for mereotopological reasoning in which two-dimensional spatial regions are treated as primitive entities. A first order predicate language ℒ with a distinguished unary predicate c(x), function-symbols +, · and - and constants 0 and 1 is defined. An interpretation ℜ for ℒ is provided in which polygonal open subsets of the real plane serve as elements of the domain. Under this interpretation the predicate c(x) is read as 'region x is connected' and the function-symbols (...) and constants are given their meaning in terms of a Boolean algebra of polygons. We give an alternative interpretation ������ base on the real closed plane which turns out to be isomorphic to ℜ. A set of axioms and a rule of inference are introduced. We prove the soundness and completeness of the calculus with respect to the given interpretation. (shrink)

Dynamic Topological Logic () is a combination of , under its topological interpretation, and the temporal logic interpreted over the natural numbers. is used to reason about properties of dynamical systems based on topological spaces. Semantics are given by dynamic topological models, which are tuples , where is a topological space, f a function on X and V a truth valuation assigning subsets of X to propositional variables. Our main result is that the set of valid formulas of over spaces (...) with continuous functions is recursively enumerable. We show this by defining alternative semantics for . Under standard semantics, is not complete for Kripke frames. However, we introduce the notion of a non-deterministic quasimodel, where the function f is replaced by a binary relation g assigning to each world multiple temporal successors. We place restrictions on the successors so that the logic remains unchanged; under these alternative semantics, becomes Kripke-complete. We then apply model-search techniques to enumerate the set of all valid formulas. (shrink)

Statistical Default Logic (SDL) is an expansion of classical (i.e., Reiter) default logic that allows us to model common inference patterns found in standard inferential statistics, e.g., hypothesis testing and the estimation of a population‘s mean, variance and proportions. This paper presents an embedding of an important subset of SDL theories, called literal statistical default theories, into stable model semantics. The embedding is designed to compute the signature set of literals that uniquely distinguishes each extension on a statistical default (...) theory at a pre-assigned error-bound probability. (shrink)

The words “content” and “character” in my title refer to the representational content and phenomenal character of color experiences. So my topic concerns the nature of our experience of color. But I will, of course, be talking about colors as well as color experience. Let me set the stage by mentioning some things, some more controversial than others, that I will be taking for granted. I assume, to begin with, that objects in the world have colors, and have them independently (...) of being perceived to have them, and independently even of there being creatures capable of perceiving them. I think, and this of course sets me apart from the many color irrealists among philosophers and color scientists, that any reasonable semantics for color terms, and any reasonable account of the reference of color concepts, should yield the result that colors are properties of external things that are realized in certain of their physical properties, namely those responsible for their reflecting or emitting the light whose impact on our retinas is involved in causing our color perceptions. This brings me to a further assumption that I shall be making, namely the truth of physicalism. I take physicalism to be the thesis that all properties of things either are or are realized in basic physical properties, where basic physical properties are the properties that underlie the behavior and causal powers of inanimate things. There are two ways in which the commitment to physicalism will figure in my discussion. First, I assume that colors are physically realized properties. Second, I assume that color experiences are physically realized. These two commitments frame the problem I am discussing – how can colors be properties realized in the microphysical properties of things, and how can color experience be so realized? I don’t think there is any generally accepted account of what it is for a property to be “realized in” other properties. For those who think, as I do, that properties are individuated by their causal features, it should seem plausible to say that property P realizes property Q just in case the forward looking causal features of Q are a subset of the forward looking causal features of P, and the backward looking.... (shrink)

Quine’s most important charge against second-, and more generally, higher-order logic is that it carries massive existential commitments. The force of this charge does not depend upon Quine’s questionable assimilation of second-order logic to set theory. Even if we take second-order variables to range over properties, rather than sets, the charge remains in force, as long as properties are individuated purely extensionally. I argue that if we interpret them as ranging over properties more reasonably construed, in accordance with an abundant (...) or deflationary conception, Quine’s charge can be resisted. This interpretation need not be seen as precluding the use of model-theoretic semantics for second-order languages; but it will preclude the use of the standard semantics, along with the more general Henkin semantics, of which it is a special case. To that extent, the approach I recommend has revisionary implications which some may find unpalatable; it is, however, compatible with the quite different special case in which the second-order variables are taken to range over definable subsets of the first-order domain, and with respect to such a semantics, some important metalogical results obtainable under the standard semantics may still be obtained. In my final section, I discuss the relations between second-order logic, interpreted as I recommend, and a strong version of schematic ancestral logic promoted in recent work by Richard Heck. I argue that while there is an interpretation on which Heck’s logic can be contrasted with second-order logic as standardly interpreted, when it is so interpreted, its differences from the more modest form of second-order logic I advocate are much less substantial, and may be largely presentational. (shrink)

In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear'” and “completely definite,”' many of the statements of analysis and set theory are “inherently vague'” and “indefinite.”' I critique his four central arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to (...) this final, remaining point will be that the concept of “being clear enough to secure definiteness” is about as clear a case of an inherently vague and indefinite concept as one might find, and as such it can bear little weight in making a case against the definiteness of analysis and set theory. (shrink)

Niche construction theory highlights the active role of organisms in modifying their environment. A subset of these modifications is the developmental niche, which concerns ecological, epistemic, social and symbolic legacies inherited by organisms as resources that scaffold their developmental processes. Since in this theory development is a situated process that takes place in a culturally structured environment, we may reasonably ask if implicit cultural biases may, in some cases, be responsible for maladaptive developmental niches. In this paper we wish (...) to argue for an affirmative answer. In order to do so, we first propose to conceptualize implicit bias as embodied perceptual habits, and then proceed to show that these habits are at least partially responsible for maladaptive developmental niches of children with Down syndrome and autism. With this framework we thus hope to bring together two fields of research that haven’t been explicitly connected: implicit bias and niche construction theory. Linking these two theories may bring benefits both to implicit bias researchers, who can extend this concept to characterize other sets of processes as biased, as well as to niche construction theorists, who will have a useful theoretical tool to diagnose maladaptive features of niches brought about by sociocultural biases. (shrink)

Our recent paper reported that 43% of reviewer comment sets shared with authors contained at least one unprofessional comment or an incomplete, inaccurate of unsubstantiated critique. Publication of this work sparked an online conversation surrounding professionalism in peer review. We collected and analyzed these social media comments as they offered real-time responses to our work and provided insight into the views held by commenters and potential peer-reviewers that would be difficult to quantify using existing empirical tools. Overall, 75% of comments (...) were positive, of which 59% were supportive and 16% shared similar personal experiences. However, a subset of negative comments emerged, that provided potential insight into the reasons underlying unprofessional comments were made during the peer-review process. These comments were classified into three main themes: forced niceness will adversely impact the peer-review process and allow for publication of poor-quality science ; dismissing comments as not offensive to another person because they were not deemed personally offensive to the reader ; and authors brought unprofessional comments upon themselves as they submitted substandard work. Here, we argue against these themes as justifications for directing unprofessional comments towards authors during the peer review process. We argue that it is possible to be both critical and professional, and that no author deserves to be the recipient of demeaning ad hominem attacks regardless of supposed provocation. Suggesting otherwise only serves to propagate a toxic culture within peer review. While we previously postulated that establishing a peer-reviewer code of conduct could help improve the peer-review system, we now posit that priority should be given to repairing the negative cultural zeitgeist that exists in peer-review. (shrink)

A well-known result, going back to the twenties, states that, under some reasonable assumptions, any logic can be characterized as the set of formulas satisfied by a matrix 〈,F〉, whereis an algebra of the appropriate type, andFa subset of the domain of, called the set of designated elements. In particular, every quasi-classical modal logic—a set of modal formulas, containing the smallest classical modal logicE, which is closed under the inference rules of substitution and modus ponens—is characterized by such a (...) matrix, wherenow is a modal algebra, andFis a filter of. If the modal logic is in fact normal, then we can do away with the filter; we can study normal modal logics in the setting of varieties of modal algebras. This point of view was adopted already quite explicitly in McKinsey and Tarski [8]. The observation that the lattice of normal modal logics is dually isomorphic to the lattice of subvarieties of a variety of modal algebras paved the road for an algebraic study of normal modal logics. The algebraic approach made available some general results from Universal Algebra, notably those obtained by Jónsson [6], and thereby was able to contribute new insights in the realm of normal modal logics [2], [3], [4], [10].The requirement that a modal logic be normal is rather a severe one, however, and many of the systems which have been considered in the literature do not meet it. For instance, of the five celebrated modal systems, S1–S5, introduced by Lewis, S4 and S5 are the only normal ones, while only SI fails to be quasi-classical. The purpose of this paper is to generalize the algebraic approach so as to be applicable not just to normal modal logics, but to quasi-classical modal logics in general. (shrink)

Previously, both Soare and Simpson considered sets without subsets of higher -degree. Cintioli and Silvestri, for a reducibility , define the concept of a -introimmune set. For the most common reducibilities , a set does not contain subsets of higher -degree if and only if it is -introimmune. In this paper we consider -introimmune and -introimmune sets and examine how structurally easy such sets can be. In other words we ask, What is the smallest class of the Kleene's Hierarchy containing (...) -introimmune sets for ? We answer the question by proving the existence of -introimmune sets in the class , bi--introimmune sets in , and bi--introimmune sets in. (shrink)